Using convex duality techniques to solve optimal control problems for diffusion processes is of course not new. It can be traced back at least to Fleming and Vermes [1]} where the philosophy is very close to ours. In Tan and Touzi, [2]} the authors extend the usual Monge-Kantorovitch optimal transportation problem to a stochastic framework. The mass is transported along a continuous semimartingale and the initial and terminal distributions are prescribed. Studying optimal control problems for the Fokker-Planck equation in order to understand the stochastic control problem is less common and it seems adapted to problems where the constraints only act on the law of the process. We refer to the works of Blaquière [3]} and more recently, Mikami [4]} and Mikami and Thieullen [5]} where similar approaches are developed in connection with the so-called Schrödinger problem. This approach has been followed recently by Guo, Loeper and Wang [6]} and Guo, Loper, Langrené and Ning [7]} for problems with various expectation constraints. In both paper, the authors show that their original problem is in duality with a problem of optimal control of sub-solutions of an HJB equation. This dual problem is solved numerically. Usually probabilists prefer to study another relaxation of the initial problem through the martingale problem (see Stroock and Varadhan [8]}), as in El Karoui, Jeanblanc-Picqué and Nguyen [9]} or Lacker [10]}. These different ways to relax the initial problem are, of course, connected and the correspondences between the diffusion processes, the martingale problem and the Fokker-Planck equation are now well established starting from the seminal work of [8]} and more recently Figalli [12]} and Trevisan [13]}.
